# Finitely Generated Abelian Groups

• Combinatorics
Hence the kernel of f_i consists of all elements divisible by p^i. Now, this is the same as saying that the kernel consists of all elements divisible by p^i for all i, which is precisely what I wrote in the equation.
Combinatorics

## Homework Statement

Let $$p$$ be prime and let $$b_1 ,...,b_k$$ be non-negative integers. Show that if :
$$G \simeq (\mathbb{Z} / p )^{b_1} \oplus ... \oplus (\mathbb{Z} / p^k ) ^ {b_k}$$
then the integers $$b_i$$ are uniquely determined by G . (Hint: consider the kernel of the homomorphism $$f_i :G \to G$$ that is multiplication by $$p^i$$ . Show that $$f_1 , f_2$$ determine $$b_1$$.
Proceed similarly )

## Homework Equations

The question, together with the theorem it should help me prove is attached (this question should help me prove the uniqueness part of the theorem)

## The Attempt at a Solution

I've tried using the hint, and considering the homomorphism $$f_1 : G \to G$$ that is defined by $$f_1 (x ) = px$$ . Its kernel is $$ker f_1 = (\mathbb{Z}_p ) ^{b_1} \bigoplus (\mathbb{Z}_p) ^ {b_2} ...$$.
But how does it help me? Am I right in my calculation of the kernel? Hope someone will be able to help me

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The hint is essentially leading you to consider a proof by induction on k. Notice that $$\ker f_1 = (\mathbb Z/p)^{b_1} \oplus (\mathbb Z/p)^{b_2} \oplus (\mathbb Z/p^2)^{b_3} \oplus \cdots = (\mathbb Z/p)^{b_1+b_2} \oplus (\mathbb Z/p^2)^{b_3} \oplus \cdots.$$

morphism said:
The hint is essentially leading you to consider a proof by induction on k. Notice that $$\ker f_1 = (\mathbb Z/p)^{b_1} \oplus (\mathbb Z/p)^{b_2} \oplus (\mathbb Z/p^2)^{b_3} \oplus \cdots = (\mathbb Z/p)^{b_1+b_2} \oplus (\mathbb Z/p^2)^{b_3} \oplus \cdots.$$

Can you please explain how did you get that this is the kernel?

Thanks a lot !

Well, if px=0 mod p^i, then p^i divides px, hence p^{i-1} divides x, i.e. x=0 mod p^i.

## What is a finitely generated abelian group?

A finitely generated abelian group is a mathematical structure consisting of a set of elements and an operation that combines any two elements to form a third element. The group is considered abelian if the operation is commutative, meaning that the order in which the elements are combined does not affect the result. Finitely generated means that the group can be generated by a finite number of elements.

## What are the main properties of finitely generated abelian groups?

The main properties of finitely generated abelian groups include finiteness, abelianness, and the existence of a finite basis. This means that the group has a finite number of elements, the operation is commutative, and the group can be generated by a finite number of elements. Additionally, finitely generated abelian groups are isomorphic to direct products of cyclic groups.

## What is the structure theorem for finitely generated abelian groups?

The structure theorem for finitely generated abelian groups states that every finitely generated abelian group is isomorphic to a direct product of cyclic groups. This means that any finitely generated abelian group can be broken down into simpler, cyclic groups.

## How are finitely generated abelian groups used in mathematics?

Finitely generated abelian groups are used in various areas of mathematics, including number theory, algebra, and topology. They are particularly important in the study of finite fields, where they are used to construct error-correcting codes. They also have applications in cryptography, where they are used to generate public and private keys.

## What are some examples of finitely generated abelian groups?

Some examples of finitely generated abelian groups include the integers under addition, the rational numbers under addition, and the real numbers under addition. Other examples include finite cyclic groups and the group of matrices with integer entries and determinant 1.

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